Abstract Suppose that G is a finite solvable group, V is a finite faithful completely reducible G -module over a field of characteristic p . In this paper, we first give explicit bounds for the degrees of the irreducible characters of G in terms of | V | in the two cases where $$3\not \mid |G|$$ 3 ∤ | G | or where the semidirect product GV has abelian Sylow 2-subgroups, respectively. We then use these results to study a conjecture of Navarro.

Abstract The Euler--Poincaré equations, firstly introduced by Henri Poincaré in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various frameworks, such as semidirect products, advected parameters, and field theory, with broad applications across physics and engineering. In this paper, we develop a discrete variational framework by establishing the discrete Euler--Poincaré reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics. The reduction is achieved through the use of group difference maps, specifically those defined using the Cayley transform or the matrix exponential. Furthermore, we extend the Kelvin--Noether theorem and its discrete analogue to characterize the Kelvin--Noether quantities for the resulting continuous and discrete Euler--Poincaré equations. As a specific application, we derive both the continuous and discrete Euler--Poincaré equations for the dynamics of underwater vehicles. Numerical simulations validate the proposed discrete scheme, demonstrating excellent long-term preservation of both total energy and Kelvin--Noether quantities. These results highlight the potential of the framework for high-fidelity simulation, control, and navigation of underwater vehicles and other complex mechanical systems.

We study the metaplectic representation of the Jacobi group (the semi-direct product of the Heisenberg group by SU(1, 1)) by using the complex Weyl correspondence. In particular, we give explicit formulas for the complexWeyl symbols of the metaplectic representation operators and we prove that the complex Weyl correspondence is a Stratonovich–Weyl correspondence for the metaplectic representation.

We propose protocols to implement non-Clifford logical gates between stabilizer codes by entangling into a non-Abelian topological order as an intermediate step. Generalizing previous approaches, we provide a framework that generates a large class of non-Clifford and nondiagonal logical gates between qudit surface codes by gauging the topological symmetry of symmetry-enriched topological orders. As our main example, we concretely detail a protocol that utilizes the quantum double of S 3 to generate a controlled-charge-conjugation ( C C ) gate between a qubit and qutrit surface code. Both the preparation of non-Abelian states and logical state injection between the Abelian and non-Abelian codes are executed via finite-depth quantum circuits with measurement and feedforward. We discuss aspects of the fault tolerance of our protocol, presenting insights into how to construct a heralded decoder for the quantum double of S 3 . We also outline how analogous protocols can be used to obtain logical gates between qudit surface codes by entangling into D ( G ) , where G is a semidirect product of Abelian groups. This work serves as a step toward classifying the computational power of non-Abelian quantum phases beyond the paradigm of anyon braiding on near-term quantum devices.

Norm of $$L^p$$-Gabor Transform on Semi-Direct Products of Locally Compact Groups

Abstract We reconsider velocity addition/subtraction in Special Relativity (SR) and re-derive its well-known non-commutative and non-associative algebraic properties in a self contained way, including various explicit expressions for the Thomas angle, the derivation of which will be seen to be not as challenging as often suggested. All this is based on the polar-decomposition theorem in the traditional component language, in which Lorentz transformations are ordinary matrices. In the second part of this paper we offer a less familiar alternative geometric view, that leads to an invariant definition of the concept of relative velocity between two states of motion, which is based on the boost-link-theorem, of which we also offer an elementary proof that does not seem to be widely known in the relativity literature. Finally we compare this to the corresponding geometric definitions in Galilei-Newton spacetime, emphasising similarities and differences. Regarding the presentation of the material we will pursue an uncompromising pedagogical strategy, willingly accepting repetitions and occasional redundancies if deemed beneficial for clarity and the avoidance of anticipated misunderstandings. An appendix with four sections includes some mathematical details on results needed in the main text, as well some recollections on notions like semi-direct products of groups and affine spaces.

Quasi-Invariant States for Actions of Semidirect Product Groups

Learning with errors over group rings constructed by semi-direct product

Abstract Cryptographic group actions have gained significant attention in recent years for their application on post-quantum Sigma protocols and digital signatures. In NIST’s recent additional call for post-quantum signatures, three relevant proposals are based on group actions: LESS, MEDS, and ALTEQ. This work explores signature optimisations leveraging a group’s factorisation. We show that if the group admits a factorisation as a semidirect product of subgroups, the group action can be restricted on a quotient space under the equivalence relation induced by the factorisation. If the relation is efficiently decidable, we show that it is possible to construct an equivalent Sigma protocol for a relationship that depends only on one of the subgroups. Moreover, if a special class of representative of the quotient space is efficiently computable via a canonical form, the restricted action is effective and does not incur in security loss. Finally, we apply these techniques to the group actions underlying LESS and MEDS, showing how they will affect the length of signatures and public keys.

It is shown that a decomposition of a complex Lie group G G into a semidirect product generates a decomposition of the algebra of analytic functionals, A ( G ) \mathscr {A}(G) , into an analytic smash product in the sense of Pirkovskii. Also, sufficient conditions are found for a semidirect product to generate similar decompositions of certain Arens–Michael completions of A ( G ) \mathscr {A}(G) . The main result: if G G is connected, then its linearization admits a decomposition into an iterated semidirect product (with the composition series consisting of Abelian factors and a semisimple factor) that induces a decomposition of algebras in a class of completions of A ( G ) \mathscr {A}(G) into iterated analytic smash products. For the extreme cases, specifically, the envelope of A ( G ) \mathscr {A}(G) in the class of all Banach algebras (also known as the Arens–Michael envelope) and the envelope in the class of Banach PI-algebras (a new concept that is introduced in this article), decompositions are presented into iterated analytic smash products.

For a finitely generated LEF group Gamma , we study the orders of finite groups admitting local embeddings of balls in a word metric on Gamma , as measured by the LEF growth function. We prove that any sufficiently smooth increasing function between n! and exp (exp (n)) is close to the LEF growth function of some finitely generated group. This is achieved by estimating the LEF growth of some semidirect products of the form {{,textrm{FSym},}}(Omega ) rtimes Gamma , where Omega curvearrowleft Gamma is an appropriate transitive action and {{,textrm{FSym},}}(Omega ) is the group of finitely supported permutations of Omega . A key tool in the proof is to identify sequences of finitely presented subgroups with short “relative” presentations. In a similar vein, we also obtain estimates on the LEF growth of some groups of the form E_{Omega } (R) rtimes Gamma , for R an appropriate unital ring and E_{Omega } (R) the subgroup of {{,textrm{Aut},}}_R (R[Omega ]) generated by all transvections with respect to basis Omega .

We derive variational integrators for stochastic Hamiltonian systems on Lie groups using a discrete version of the stochastic Hamiltonian phase space principle. The structure-preserving properties of the resulting scheme, such as symplecticity, preservation of the Lie-Poisson structure, preservation of the coadjoint orbits, and preservation of Casimir functions, are discussed, along with a discrete Noether theorem for subgroup symmetries. We also consider in detail the case of stochastic Hamiltonian systems with advected quantities, studying the associated structure-preserving properties in relation to semidirect product Lie groups. A full convergence proof for the scheme is provided for the case of the Lie group of rotations. Several numerical examples are presented, including simulations of the free rigid body and the heavy top.

A D-derivation of a Lie algebra L is a linear map φ for which there exists a derivation D such that φ([x,y])=[φ(x),y]+[x,D(y)] for all x,y∈L. This paper presents explicit structural results concerning D-derivations in Lie algebras over arbitrary fields. It is established that the set of D-derivations forms a Lie algebra, which decomposes as the sum of derivations and centroids, intersecting precisely at the space of central derivations. For centerless Lie algebras, the inclusion chain for D-derivations within existing derivation classes is completed, resulting in a refined hierarchy. It is proven that for both perfect and centerless Lie algebras, D-derivations decompose as a direct sum of derivations and centroids. In particular, for semisimple Lie algebras, it is shown that DerD(L)=ad(L)⊕C(L), and for simple Lie algebras over an algebraically closed field of characteristic zero, DerD(L)=ad(L)⊕FidL. Furthermore, for any centerless Lie algebra, the Lie algebra of D-derivations is shown to be isomorphic to the semidirect product of the derivation and centroid algebras, with explicit descriptions provided for semisimple and solvable cases. Examples involving so(3), so(1,3), aff(1), and h3 confirm these decompositions and offer matrix realizations of their D-derivations, thereby supporting and illustrating the main theorems.

A bstract We revisit the holographic symmetry algebra in the MHV sector. We find an infinite dimensional Abelian symmetry algebra whose generators are the conformally soft negative helicity gravitons and gluons. So the complete symmetry algebra in the MHV graviton sector is a semideirect product of the w 1+ ∞ algebra and the infinite dimensional Abelian algebra. Similarly in the MHV gluon sector the symmetry algebra is a semidirect product of the S algebra and the infinite dimensional Abelian algebra. The extended symmetry algebra has some use. For example, it is known for sometime that an n point MHV amplitude satisfies ( n − 2) Knizhnik-Zamolodchikov (KZ) type equations. So two equations are missing. We show that the extended symmetry algebra has additional null states whose decoupling give rise to the two missing equations.

In this paper, we study the irreducible representations of skew left braces of size pq, which is equivalent to studying the representation theory of groups of order p 2 q 2 arising from skew left braces, where p < q are primes. To achieve this, we classify all semidirect product groups Λ A associated with skew left braces A of order pq, up to isomorphism.

The classification of the unitary irreducible representations of symmetry groups is a cornerstone of modern quantum physics, as it provides the fundamental building blocks for constructing the Hilbert spaces of theories admitting these symmetries. In the context of gravitational theories, several arguments point toward the existence of a universal symmetry group associated with corners, whose structure is the same for every diffeomorphism-invariant theory in any dimension. Recently, the representations of the maximal central extension of this group in the two-dimensional case have been classified using purely algebraic techniques. In this work, we present a complementary and independent derivation based on Kirillov’s orbit method. We study the coadjoint orbits of the group SL ( 2 , R ) ˜ ⋉ H 3 , where H 3 is the Heisenberg group of a quantum particle in one dimension. Our main result is that, despite the non-Abelian nature of the normal subgroup in the semidirect product, these orbits admit a simple description. In a coordinate system associated with modified Lie algebra generators, the orbits factorize into a product of coadjoint orbits of SL ( 2 , R ) and H 3 . The subsequent geometric quantization of these factorized orbits successfully reproduces the known representations.

Abstract This paper numerically investigates Euler–Poincaré equations arising from a self-semidirect product group structure. A monolithic energy-preserving continuous Galerkin finite element method is used to study geodesic equations associated with the semidirect product of the diffeomorphism group on a circle with itself. Theoretically predicted peakon solutions are observed as an emergent behaviour. In addition, complicated nonlinear transfers of energy are associated with the semidirect product coupling, where amongst various nonlinear interactions, we observe coupled peakon behaviour. We show that the observed peakon-locking behaviour is expected analytically as an admissible singular solution of the dynamical system. A mimetic (C-grid) finite difference method is used to study the geodesic flow of the semidirect product of the volume preserving diffeomorphism group with itself, where similar coupling behaviour is observed in the vorticity variables. We also investigate coadjoint and Lie–Poisson structures in the context of geodesic equations on semidirect product groups, where the underlying group is first extended by central extension or semidirect product.

We study extended associative semigroups (briefly, EAS), an algebraic structure used to define generalizations of the operad of associative algebras, and the subclass of commutative extended diassociative semigroups (briefly, CEDS), which are used to define generalizations of the operad of pre-Lie algebras. We give families of examples based on semigroups or on groups, as well as a classification of EAS of cardinality two. We then define linear extended associative semigroups as linear maps satisfying a variation of the braid equation. We explore links between linear EAS and bialgebras and Hopf algebras. We also study the structure of non-degenerate finite CEDS and show that they are obtained by semi-direct and direct products involving two groups.

Abstract A group G has the endomorphism kernel property (EKP) if every congruence relation θ on G is the kernel of an endomorphism on G . In this note we fully characterize extraspecial groups which have EKP and show two series of special groups which have EKP, these series are specific semidirect products of types C p 2 n ⋊ C p and C p × C p n ⋊ C p $\left(C_{p^2}{ }^n\right) \rtimes C_p \text { and }\left(C_p \times C_p\right)^n \rtimes C_p$ for all primes p and all n .

A locally conformally product (LCP) structure on a compact conformal manifold is a closed non-exact Weyl connection (i.e. a linear connection which is locally but not globally the Levi-Civita connection of Riemannian metrics in the conformal class), with reducible holonomy. A left-invariant LCP structure on a compact quotient [Formula: see text] of a simply connected Riemannian Lie group [Formula: see text] with Lie algebra [Formula: see text] can be characterized in terms of a closed one-form [Formula: see text] and a non-zero subspace [Formula: see text] satisfying some algebraic conditions. We show that these conditions are equivalent to the fact that [Formula: see text] is isomorphic to a semidirect product of a non-unimodular Lie algebra acting on an abelian one by a conformal representation. This extends to the general case results from Andrada et al. [1] holding for solvmanifolds. In addition, we construct explicit examples of compact LCP manifolds which are not solvmanifolds.